5. State-Space Search: State Spaces

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Foundations of Artificial Intelligence

5. State-Space Search: State Spaces

Malte Helmert

University of Basel

March 8, 2021

M. Helmert (University of Basel) Foundations of Artificial Intelligence March 8, 2021 1 / 22

Foundations of Artificial Intelligence

March 8, 2021 — 5. State-Space Search: State Spaces

5.1 State-Space Search Problems 5.2 Formalization

5.3 State-Space Search 5.4 Summary

5. State-Space Search: State Spaces State-Space Search Problems

5.1 State-Space Search Problems

Classical State-Space Search Problems Informally

(Classical) state-space search problems are among the “simplest”

and most important classes of AI problems.

objective of the agent:

I from a given initial state I apply a sequence of actions I in order to reach a goal state

performance measure: minimize total action cost

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Motivating Example: 15-Puzzle

9 2 12 6

5 7 14 13

3 1 11

15 4 10 8

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15

Classical Assumptions

“classical” assumptions:

I no other agents in the environment (single-agent) I always knows state of the world (fully observable) I state only changed by the agent (static)

I finite number of states/actions (in particular discrete) I actions have deterministic effect on the state

can all be generalized (but not in this part of the course) For simplicity, we omit “classical” in the following.

Classification

Classification:

State-Space Search environment:

I static vs. dynamic

I deterministic vs. non-deterministic vs. stochastic I fully vs. partially vs. not observable

I discrete vs. continuous I single-agent vs. multi-agent problem solving method:

I problem-specific vs. general vs. learning

Search Problem Examples

I toy problems: combinatorial puzzles

(Rubik’s Cube, 15-puzzle, towers of Hanoi, . . . ) I scheduling of events, flights, manufacturing tasks I query optimization in databases

I behavior of NPCs in computer games I code optimization in compilers I verification of soft- and hardware I sequence alignment in bioinformatics I route planning (e.g., Google Maps) I . . .

thousands of practical examples

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State-Space Search: Overview

Chapter overview: state-space search I 5.–7. Foundations

I 5. State Spaces

I 6. Representation of State Spaces I 7. Examples of State Spaces I 8.–12. Basic Algorithms I 13.–19. Heuristic Algorithms

5. State-Space Search: State Spaces Formalization

5.2 Formalization

Formalization

preliminary remarks:

I to cleanly study search problems we need a formal model I fundamental concept: state spaces

I state spaces are (labeled, directed) graphs I paths to goal states represent solutions I shortest paths correspond to optimal solutions

State Spaces: Example

State spaces are often depicted as directed graphs.

I states: graph vertices I transitions: labeled arcs

(here: colors instead of labels) I initial state: incoming arrow I goal states: marked

(here: by the dashed ellipse) I actions: the arc labels

I action costs: described separately (or implicitly = 1)

A B C

D

E F

initial state

goal states

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State Spaces

Definition (state space)

A state space or transition system is a 6-tuple S = hS , A, cost, T , s ₀ , S _? i with

I S: finite set of states I A: finite set of actions I cost : A → R ⁺ ₀ action costs

I T ⊆ S × A × S transition relation; deterministic in hs, ai (see next slide)

I s ₀ ∈ S initial state I S _? ⊆ S set of goal states

German: Zustandsraum, Transitionssystem, Zust¨ ande, Aktionen, Aktionskosten, Transitions-/ ¨ Ubergangsrelation, deterministisch, Anfangszustand, Zielzust¨ ande

State Spaces: Transitions, Determinism

Definition (transition, deterministic)

Let S = hS, A, cost, T , s ₀ , S _? i be a state space.

The triples hs , a, s ⁰ i ∈ T are called (state) transitions.

We say S has the transition hs , a, s ⁰ i if hs, a, s ⁰ i ∈ T . We write this as s − → ^a s ⁰ , or s → s ⁰ when a does not matter.

Transitions are deterministic in hs , ai: it is forbidden to have both s − → ^a s ₁ and s − → ^a s ₂ with s ₁ 6= s ₂ .

State Spaces: Terminology

We use common terminology from graph theory.

Definition (predecessor, successor, applicable action) Let S = hS, A, cost, T , s ₀ , S _? i be a state space.

Let s, s ⁰ ∈ S be states with s → s ⁰ . I s is a predecessor of s ⁰

I s ⁰ is a successor of s

If s − → ^a s ⁰ , then action a is applicable in s.

German: Vorg¨ anger, Nachfolger, anwendbar

State Spaces: Terminology

We use common terminology from graph theory.

Definition (path)

Let S = hS, A, cost, T , s ₀ , S _? i be a state space.

Let s ⁽⁰⁾ , . . . , s ⁽ⁿ⁾ ∈ S be states and π ₁ , . . . , π _n ∈ A be actions such that s ⁽⁰⁾ −→ ^π

¹

s ⁽¹⁾ , . . . , s ⁽ⁿ⁻¹⁾ −→ ^π

ⁿ

s ⁽ⁿ⁾ .

I π = hπ ₁ , . . . , π _n i is a path from s ⁽⁰⁾ to s ⁽ⁿ⁾ I length of π: |π| = n

I cost of π: cost(π) = P n

i =1 cost(π _i ) German: Pfad, L¨ ange, Kosten

I paths may have length 0

I sometimes “path” is used for state sequence hs ⁽⁰⁾ , . . . , s ⁽ⁿ⁾ i

or sequence hs ⁽⁰⁾ , π , s ⁽¹⁾ , . . . , s ⁽ⁿ⁻¹⁾ , π , s ⁽ⁿ⁾ i

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State Spaces: Terminology

more terminology:

Definition (reachable, solution, optimal) Let S = hS, A, cost, T , s ₀ , S _? i be a state space.

I state s is reachable if a path from s ₀ to s exists I paths from s ∈ S to some state s _? ∈ S _?

are solutions for/from s

I solutions for s ₀ are called solutions for S I optimal solutions (for s ) have minimal costs

among all solutions (for s)

German: erreichbar, L¨ osung von/f¨ ur s , optimale L¨ osung

5. State-Space Search: State Spaces State-Space Search

5.3 State-Space Search

State-Space Search

State-space search is the algorithmic problem of finding solutions in state spaces or proving that no solution exists.

In optimal state-space search, only optimal solutions may be returned.

German: Zustandsraumsuche, optimale Zustandsraumsuche

Learning Objectives for State-Space Search

Learning Objectives for the Topic of State-Space Search I understanding state-space search:

What is the problem and how can we formalize it?

I evaluate search algorithms:

completeness, optimality, time/space complexity I get to know search algorithms:

uninformed vs. informed; tree and graph search I evaluate heuristics for search algorithms:

goal-awareness, safety, admissibility, consistency I efficient implementation of search algorithms I experimental evaluation of search algorithms

I design and comparison of heuristics for search algorithms

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5. State-Space Search: State Spaces Summary

5.4 Summary

5. State-Space Search: State Spaces Summary

Summary

I classical state-space search problems:

find action sequence from initial state to a goal state I performance measure: sum of action costs

I formalization via state spaces:

I states, actions, action costs, transitions, initial state, goal states

I terminology for transitions, paths, solutions I definition of (optimal) state-space search

5. State-Space Search: State Spaces

Foundations of Artificial Intelligence

5. State-Space Search: State Spaces

Malte Helmert

University of Basel

March 8, 2021

Foundations of Artificial Intelligence

March 8, 2021 — 5. State-Space Search: State Spaces

5.1 State-Space Search Problems 5.2 Formalization

5.3 State-Space Search 5.4 Summary

5.1 State-Space Search Problems

Classical State-Space Search Problems Informally

(Classical) state-space search problems are among the “simplest”

and most important classes of AI problems.

objective of the agent:

I from a given initial state I apply a sequence of actions I in order to reach a goal state

performance measure: minimize total action cost

Motivating Example: 15-Puzzle

9 2 12 6

5 7 14 13

3 1 11

15 4 10 8

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15

Classical Assumptions

“classical” assumptions:

I no other agents in the environment (single-agent) I always knows state of the world (fully observable) I state only changed by the agent (static)

I finite number of states/actions (in particular discrete) I actions have deterministic effect on the state

can all be generalized (but not in this part of the course) For simplicity, we omit “classical” in the following.

Classification

Classification:

State-Space Search environment:

I static vs. dynamic

I deterministic vs. non-deterministic vs. stochastic I fully vs. partially vs. not observable

I discrete vs. continuous I single-agent vs. multi-agent problem solving method:

I problem-specific vs. general vs. learning

Search Problem Examples

I toy problems: combinatorial puzzles

(Rubik’s Cube, 15-puzzle, towers of Hanoi, . . . ) I scheduling of events, flights, manufacturing tasks I query optimization in databases

I behavior of NPCs in computer games I code optimization in compilers I verification of soft- and hardware I sequence alignment in bioinformatics I route planning (e.g., Google Maps) I . . .

thousands of practical examples

State-Space Search: Overview

Chapter overview: state-space search I 5.–7. Foundations

I 5. State Spaces

I 6. Representation of State Spaces I 7. Examples of State Spaces I 8.–12. Basic Algorithms I 13.–19. Heuristic Algorithms

5.2 Formalization

Formalization

preliminary remarks:

I to cleanly study search problems we need a formal model I fundamental concept: state spaces

I state spaces are (labeled, directed) graphs I paths to goal states represent solutions I shortest paths correspond to optimal solutions

State Spaces: Example

State spaces are often depicted as directed graphs.

I states: graph vertices I transitions: labeled arcs

(here: colors instead of labels) I initial state: incoming arrow I goal states: marked

(here: by the dashed ellipse) I actions: the arc labels

I action costs: described separately (or implicitly = 1)

A B C

D

E F

initial state

goal states

State Spaces

Definition (state space)

A state space or transition system is a 6-tuple S = hS , A, cost, T , s 0 , S ? i with

I S: finite set of states I A: finite set of actions I cost : A → R + 0 action costs

I T ⊆ S × A × S transition relation; deterministic in hs, ai (see next slide)

I s 0 ∈ S initial state I S ? ⊆ S set of goal states

German: Zustandsraum, Transitionssystem, Zust¨ ande, Aktionen, Aktionskosten, Transitions-/ ¨ Ubergangsrelation, deterministisch, Anfangszustand, Zielzust¨ ande

State Spaces: Transitions, Determinism

Definition (transition, deterministic)

Let S = hS, A, cost, T , s 0 , S ? i be a state space.

The triples hs , a, s 0 i ∈ T are called (state) transitions.

We say S has the transition hs , a, s 0 i if hs, a, s 0 i ∈ T . We write this as s − → a s 0 , or s → s 0 when a does not matter.

Transitions are deterministic in hs , ai: it is forbidden to have both s − → a s 1 and s − → a s 2 with s 1 6= s 2 .

State Spaces: Terminology

We use common terminology from graph theory.

Definition (predecessor, successor, applicable action) Let S = hS, A, cost, T , s 0 , S ? i be a state space.

Let s, s 0 ∈ S be states with s → s 0 . I s is a predecessor of s 0

A state space or transition system is a 6-tuple S = hS , A, cost, T , s ₀ , S _? i with

I S: finite set of states I A: finite set of actions I cost : A → R ⁺ ₀ action costs

I s ₀ ∈ S initial state I S _? ⊆ S set of goal states

Let S = hS, A, cost, T , s ₀ , S _? i be a state space.

The triples hs , a, s ⁰ i ∈ T are called (state) transitions.

We say S has the transition hs , a, s ⁰ i if hs, a, s ⁰ i ∈ T . We write this as s − → ^a s ⁰ , or s → s ⁰ when a does not matter.

Transitions are deterministic in hs , ai: it is forbidden to have both s − → ^a s ₁ and s − → ^a s ₂ with s ₁ 6= s ₂ .

Definition (predecessor, successor, applicable action) Let S = hS, A, cost, T , s ₀ , S _? i be a state space.

Let s, s ⁰ ∈ S be states with s → s ⁰ . I s is a predecessor of s ⁰

I s ⁰ is a successor of s

If s − → ^a s ⁰ , then action a is applicable in s.

Let S = hS, A, cost, T , s ₀ , S _? i be a state space.

Let s ⁽⁰⁾ , . . . , s ⁽ⁿ⁾ ∈ S be states and π ₁ , . . . , π _n ∈ A be actions such that s ⁽⁰⁾ −→ ^π

s ⁽¹⁾ , . . . , s ⁽ⁿ⁻¹⁾ −→ ^π

s ⁽ⁿ⁾ .

I π = hπ ₁ , . . . , π _n i is a path from s ⁽⁰⁾ to s ⁽ⁿ⁾ I length of π: |π| = n

i =1 cost(π _i ) German: Pfad, L¨ ange, Kosten

I sometimes “path” is used for state sequence hs ⁽⁰⁾ , . . . , s ⁽ⁿ⁾ i

or sequence hs ⁽⁰⁾ , π , s ⁽¹⁾ , . . . , s ⁽ⁿ⁻¹⁾ , π , s ⁽ⁿ⁾ i

Definition (reachable, solution, optimal) Let S = hS, A, cost, T , s ₀ , S _? i be a state space.

I state s is reachable if a path from s ₀ to s exists I paths from s ∈ S to some state s _? ∈ S _?

I solutions for s ₀ are called solutions for S I optimal solutions (for s ) have minimal costs